When comparing MonoGame vs Source, the Slant community recommends Source for most people. In the question“What are the best 3D game engines?”Source is ranked 15th while MonoGame is ranked 26th. The most important reason people chose Source is:

Source includes Source Filmmaker, a video capture and editing application.

Pro

Open source

All the code is available to you ensuring you'll have the ability to make changes when you need to or even port to whole new platforms.

Pro

Managed code

By leveraging C# and other .NET languages on Microsoft and Mono platforms you can write modern, fast, and reliable game code.

Pro

Performance on desktop

The performance on desktop platforms matches that of C++, but you still get all the pleasant features that C# has to offer.

Pro

Good community

The community MonoGame has to offer is helpful and mature.

Pro

Well-known and documented API

The framework implements the XNA 4 API, so games made in XNA can be ported to other platforms using this. This was the same API used by the Xbox Live Indie Games platform so there's lots of documentation online for it.

Pro

Has a built-in video capture and editing application

Source includes Source Filmmaker, a video capture and editing application.

Pro

Great documentation

Valve's official documentation website is great for newbies. It demonstrates the pros and cons of the engine (and since the website isn't maintained by Valve, but instead the community, the pros and cons are largely unbiased). There are also a number of pages dedicated to entities used within official Valve games and also community-made mods that were turned into full-fledged games by Valve. These pages explain the ins and outs of how most source programming works. There are also guides for Valve's tools which are both included in Source SDK and in any Valve-developed game.

Cons

Con

Non-Windows tools are a bit funky

Monogame support for Xamarin Studio or Monodevelop is a bit shaky especially for library references. Only good non-Windows IDE compatible with MonoGame is Rider and that costs money & isn't open-source.

Con

SDK is outdated and difficult to use

Source SDK has not been updated in ages, and has instead been "re-released" under different names, e.g "Source SDK 2013 Singleplayer".

It's honestly easier to use the version of SDK included with any Source game, namely Portal 2 or DOTA 2, since both have a variant of Source SDK that is more updated than anything you can find in the tools section of Steam.

Con

Only for mods

Normally, you can only use the Source Engine to develop "source-mods" (as Steam calls them), however the developer wiki is correct in saying Valve have a proven track-record for finding source-mods and turning them into fully-fledged games, Black Mesa Source is a good example of this, as it began life as a source-mod available for free, however Valve turned it into a fully-fledged and paid game.